On the images of ellipses under similarities

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ON THE IMAGES OF ELLIPSES UNDER

SIMILARITIES

Nihal Yılmaz ¨Ozg¨ur

Abstract

We consider ellipses corresponding to any norm function on the com-plex plane and determine their images under the similarities which are special M¨obius transformations.

1 Introduction

It is well-known that Möbius transformations map circles to circles where straight lines are considered to be circles through ∞. It is also well-known that all norms on C are equivalent. In [5], the present author considered cir-cles corresponding to any norm function and determined their images under the Möbius transformations on the complex plane. Recently, in [2] and [3], Adam Coffman and Marc Frantz considered the images of non-circular ellipses (corresponding to the Euclidean norm function) under the Möbius transforma-tions. In [6], the present author determined the images of non-circular ellipses under the harmonic Möbius transformations.

Motivated by the above studies, we consider the images of ellipses corre-sponding to any norm function on C under the M¨obius transformations.

Key Words: M¨obius transformation, ellipse, norm.

2010 Mathematics Subject Classification: Primary 30C35; Secondary 51F15. Received: February, 2012.

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Throughout the paper, we consider the real linear space structure of the complex plane C and investigate the answer of the following question:

If w = T (z) is a M¨obius transformation and k.k is any norm function on C, then does T take ellipses to ellipses in this norm?

Note that all M¨obius transformations do not map ellipses to ellipses cor-responding to the Euclidean norm function on C. From [2] and [3], we know that the M¨obius transformations which map ellipses to ellipses are similarity transformations. In our case, we see that the rotation map z → eiφz do not map ellipses to ellipses for every value of the real number φ. Thus we restrict our investigations to similarity transformations.

2 Main results

We give a brief account of M¨obius transformations (see [1] and [4] for more details).

A M¨obius transformation T is a function of the form T (z) = az + b

cz + d; a, b, c, d ∈ C and ad − bc 6= 0. (2.1) Such transformations form a group under composition. The M¨obius transfor-mations with c = 0 form the subgroup of similarities. Such transfortransfor-mations have the form

z → αz + β; α, β ∈ C, α 6= 0. (2.2) The transformation z → 1

z is called an inversion. Here we use the well-known

fact that every M¨obius transformation T of the form (2.1) is a composition of finitely many similarities and inversions.

Let k.k be any norm function on C. A circle whose center is at z0 and of

radius r is denoted by Sr(z0) and defined by Sr(z0) = {z ∈ C : kz − z0k = r}.

An ellipse is the locus of points z with the property that the sum of the dis-tances from z to two given fixed points, say F1and F2, is a constant. The two

fixed points are called foci. Thus the set {z ∈ C : kz − F1k + kz − F2k = r} is

the ellipse with foci F1 and F2. We denote this ellipse by Er(F1, F2). If the

two foci coincide, then the ellipse is a circle.

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Lemma 2.1. [5] Let k.k be any norm function on the complex plane. Then for every φ ∈ R, the following function define a norm on the complex plane:

kzk_φ = e−iφz . (2.3) We begin the following lemma.

Lemma 2.2. Let k.k be any norm on C. Then the similarity transformations of the form

f (z) = αz + β; α 6= 0, α ∈ R, (2.4) map ellipses to ellipses corresponding to this norm function.

Proof. Let k.k be any norm and let Er(F1, F2) be any ellipse corresponding

to this norm. If f (z) is a similarity transformation of the form (2.4), then the image of Er(F1, F2) under f is the ellipse E|α|r(f (F1), f (F2)). Indeed, we

have

kf (z) − f (F1)k + kf (z) − f (F2)k

= kαz + β − (αF1+ β)k + kαz + β − (αF2+ β)k

= kα(z − F1)k + kα(z − F2)k

= |α| (kz − F1k + kz − F2k) = |α| r.

Now we consider the norm functions defined in (2.3). Notice that for the Euclidean norm, all of the norm functions k.k_φ are equal to the Euclidean norm. For any other norm function we have k.k_kπ_{= k.k where k ∈ Z.}

Then we can give the following theorem:

Theorem 2.1. Let w = f (z) = αz + β; α 6= 0, α, β ∈ C. Then for every ellipse Er(F1, F2) corresponding to any norm function k.k on C, f (Er(F1, F2))

is an ellipse corresponding to the same norm function or corresponding to the norm function kzk_φ = e−iφ.z , where φ = arg(α).

Proof. Let w = f (z) = αz + β; α 6= 0, α, β ∈ C. If Er(F1, F2) is an Euclidean

ellipse, then from [3] we know that f (Er(F1, F2)) is again an Euclidean ellipse.

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|α| eiφ_{z + β; α 6= 0, φ = arg(α) and let f}

1(z) = eiφz, f2(z) = |α| z + e−iφβ.

We have f (z) = (f1◦ f2)(z).

Then by Lemma 2.2, the transformation f2(z) maps ellipses to ellipses

corresponding to this norm function. Let w = f1(z) = eiφz, φ 6= kπ, k ∈ Z.

Now we consider the norm function k.k_φ given in Lemma 2.1. We get kw − f (F1)kφ+ kw − f (F2)kφ = eiφ(z − F1) _φ+ eiφ(z − F2) _φ = e−iφeiφ(z − F1) + e−iφeiφ(z − F2) = kz − F1k + kz − F2k = r.

This shows that the image of the ellipse Er(F1, F2) under the transformation

w = f1(z) = eiφz, (φ 6= kπ, k ∈ Z) is the ellipse Er(f (F1), f (F2))

correspond-ing to the norm function k.k_φ given in (2.3).

We note that we do not know the exact values of φ for which k.k_φ= k.k. This is an open problem. If k.k_φ = k.k, then the transformation f1(z) =

eiφ_{z maps ellipses to ellipses corresponding to this norm function. In general}

f1(z) = eiφz do not map ellipses to ellipses corresponding to the same norm

function. For example, let k.k be any norm with k1k 6= kik and φ = π₂. Assume that kzkπ

2 = kzk for all z ∈ C. For z = 1 we have kik = k1k,

which is a contradiction. Therefore the transformation z → eπ2iz maps ellipses

corresponding to the norm function k.k to ellipses corresponding to the norm function k.kπ

2. We give the following conjecture for the norm functions with

the properties k1k = kik and kzk = kzk for all z ∈ C.

Conjecture 2.1. Let k.k be any norm on C with k1k = kik. Assume that kzk = kzk for all z ∈ C. Then we have k.kπ

2 = k.k and hence the

transforma-tion z → eπ2iz maps ellipses to ellipses corresponding to this norm function.

If this conjecture is true, then we have also the transformation z → eπ2iz

maps circles to circles corresponding to this norm function as a corollary. Example 2.1. Let us consider the norm function

kzk = 2 |x| + |y|

on C. Let F1= −1 and F2= 1. The image of the ellipse E6(F1, F2) under the

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ze

Π 2i_z

Figure 1:

it is the ellipse E6(−i, i) corresponding to the norm function kzkπ 2

= |x|+2 |y|, (see F igure 1).

Finally we note that Lemma 2.2 and Theorem 2.1 hold also for hyperbolas corresponding to any norm function on the complex plane.

References

[1] A. F. Beardon, Algebra and Geometry, Cambridge University Press, Cam-bridge, 2005.

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[2] A. Coffman and M. Frantz, Ellipses in the Inversive Plane, MAA Indiana Section Meeting, Mar. 2003.

[3] A. Coffman and M. Frantz, M¨obius Transformations and Ellipses, The Pi Mu Epsilon Journal, 12 (2007), no.6, 339-345.

[4] G. A. Jones and D. Singerman, Complex functions. An algebraic and geometric viewpoint, Cambridge University Press, Cambridge, 1987. [5] N. Yılmaz Özgür, On some mapping properties of Möbius

transforma-tions, Aust. J. Math. Anal. Appl., 6 (2009), no. 1, Art. 13, 8 pp.

[6] N. Yılmaz Özgür, Ellipses and Harmonic Möbius Transformations, An. S¸t. Univ. Ovidius Constanta, 18 (2010), no. 2, 201-208.

Nihal YILMAZ ÖZG ÜR, Department of Mathematics, Balıkesir University, Balıkesir, Türkiye.